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Journal
of Economic
Dynamics
and Control
12 (1988) 489-502.
North-Holland
THE CONVERGENCE OF MULTIVARIATE ‘UNIT ROOT’
DISTRIBUTIONS TO THEIR ASYMPTOTIC LIMITS
The Case of Money-Income Causality
Lars LJUNGQVIST
University
of Mmnesota, Minneapolis, MN 55455, USA
Myungsoo
Northwestern
PARK
University, Evanston, IL 60208, USA
James H. STOCK*
Harvard University, Cambridge, MA 02138, USA
Mark W. WATSON*
Northwestern
Received November
University, Evunston, IL 60208, USA
1987, final version received January
1988
We examine
the quality of recently developed
asymptotic
approximations
to the sampling
distributions
of various statistics in levels regressions when the regressors have unit roots. The
calculations
were performed
using a bivariate probability
model typical of some considered
in
applied macroeconomic
research: the parameters
of the model were obtained by estimating
a
VAR using postwar U.S. money and industrial production
growth rates, resulting in pseudo-data
that are I(1) with drifts. With 100 observations
the asymptotic approximations
are often found to
be adequate;
with 400 observations
they are generally good. In addition, when the statistics have
nonstandard
distributions,
both the asymptotic
and exact distributions
differ substantially
from
the usual normal or x2 distributions
that would apply were the regressors stationary.
1. Introduction
There has been considerable
theoretical progress towards understanding
the
asymptotic
behavior of regression statistics when some or all of the variables
are integrated.
A central objective of this research is to provide guidance in
approximating
the sampling distributions
of estimators and test statistics in
data sets in which there are one or more unit roots in the multivariate
representation
of the series. But do these asymptotic approximations
provide a
*Stock and Watson
no. SES-86-18984.
01651889/88/$3.50~1988,
thank the National
Science Foundation
Elsevier Science Publishers
for financial
B.V. (North-Holland)
support
through
grant
490
L. Ljungqoisr, Multioariare
‘unit root’ distributions
good guide to sampling distributions in sample sizes and probability models
‘typical’ of those found in empirical macroeconomic research? Previous researchers have used Monte Carlo techniques to check the validity of asymptotic approximations in related problems with moderate sample sizes; e.g.,
tests of univariate or multivariate unit roots [e.g., Dickey and Fuller (1979),
Phillips and Ouliaris (1988), and Schwert (1987)] and least squares estimators
of cointegrating vectors [e.g., Banerjee et al. (1986), Stock (1987)]. In many
contexts, however, the question of interest is different, often reducing to the
behavior of certain test statistics - such as F-tests of exclusion restrictions - in
linear time series models.
In this paper we examine the rate of convergence of the sampling distribution to the asymptotic distribution using a Monte Carlo experiment involving
two variables, both constructed to be integrated of order one. The major
difficulty in designing such an experiment is determining a probability model
to generate the data that is ‘realistic’: typical linear time series models
encountered in applied research involve multiple variables and many lags,
which in turn requires specifying many parameters in developing the experimental design. Our solution to this problem is to consider a particular linear
probability model (or ‘data generation process’) that has received widespread
attention in the macroeconomic literature: a bivariate vector autoregression
(VAR) with money and output, estimated using U.S. postwar data.’ This
model is estimated imposing the assumption that each series is integrated of
order one, that the series are not cointegrated, that each series contains a
nonzero drift, and that money does not enter the output equation. This final
assumption permits the generation of the distribution of the Granger causality
test statistic under the null hypothesis.
We examine the asymptotic and Monte Carlo distributions of four statistics
in a OLS regression of income on lags of income and money, one of which is
the usual ‘Granger causality’ F-statistic testing the hypothesis that money does
not enter the income equation. As is discussed below, three of these - including the Granger causality statistic - have nonstandard asymptotic distributions. These statistics - one point estimate and three test statistics - are
examined for two regressions: one in which the only deterministic term is a
constant and a second in which a linear time trend is added as well.’
The details of the experimental design and a brief discussion of the relevant
asymptotics are presented in section 2. The results are discussed in section 3,
and we conclude with section 4.
‘For recent reviews of the literature on the money-output relation, see Eichenbaum and
Singleton (1986), Blanchard (1987), Christiano and Ljungqvist (1987), and Stock and Watson
(1987).
2These specifications are of interest for macroeconomic as well as econometric reasons: as
Bemanke (1986) and Runkle (1987) point out, different macroeconomic conclusions can obtain
when time is added as a regressor in similar levels specifications.
491
L. Ljungqvist, Mulrivariafe ‘unit root’ distributions
2. Experimental
design
2.1. The probability
We adopt
model and asymptotic
the probability
behavior
model
in which the first through the fifth lags of Ay, and Am, appear on the
right-hand
side. The parameters
of (1) were estimated by OLS using 460
observations
on postwar U.S. monthly industrial production
and Ml growth.3
The resulting parameters are presented in the appendix.
We study four statistics computed
in two specifications
of the output
equation in the unconstrained
levels VAR(6) implied by (1). In the first, the
level of income is regressed against lagged income, lagged money, and a
constant:
Regression
C:
Regress
y, on
(1;
Yr_1,Y,_2,...3
Yt-6;
m~-l~mf-2~“.3
mt-6)’
Sims, Stock and Watson (1986) examine the asymptotic properties of OLS
regressions
when the data are generated by a probability
model such as (1).
U:ing their arguments, the estimated sum of the coefficients on lagged money
[j?l,,,(l)] will have a nonstandard
asymptotic
distribution;
so will the usual
t-ratio produced by OLS regression packages testing the (correct) hypothesis
F-test [call
that this sum is zero [call this tp,mclj]. The usual Granger-causality
m,_,)] will also have a nonstandard
limiting distribution.
this F(m,_l,...,
However, a F-test of linear restrictions on any proper subset of the coefficients
will have the usual asymptotic x2 distribution,
since such restrictions can be
3The entire industrial production
(TF’) series and the Ml data since 1959 were taken from the
Citibase data base. The pre-1959 Ml data are taken from Christian0 and Ljungqvist. The IP data
were transformed
by taking first differences of their logarithm [so that Ay, = A log( IP,)]. The
transformation
applied to Ml was somewhat more involved, since money growth appears to
contain a linear time trend over this sample period [see Stock and Watson (1987)]. Accordingly,
money growth was first detrended by taking the residuals of a regression (from 1948:l to 1985:12)
of A log(MI,)
against a constant and time. Since this series has mean zero by construction,
the
average postwar
drift in money was restored to the data by adding the postwar average of
A log( Ml,) to these residuals. These transformed
data were used to estimate the constrained
VAR(5) in (1) over 1948:7 to 1985:12, using earlier observations
as initial values in the
regressions.
L. Ljungquist,Multiuuriute
492
‘unit root’ distributions
rewritten as restrictions
on mean zero, stationary variables. In particular,
this
will be true of the F-test of the hypothesis that the coefficients on all lags of
money except the first are zero [call this F(m,_,,..
. . m,_,)]. We study the
quality
of the asymptotic
approximation
to the sampling
distribution
by
considering
these four statistics.4
Because both variables contain nonzero drifts, y, is in effect ‘detrending’ m,
in regression
C. Since the resultant
‘detrended’
series is integrated,
this
detrending
affects the asymptotic
distribution
of the first three statistics.
Consequently,
we also consider
the quality of the approximation
of the
asymptotic
distribution
when a linear time trend is included as a regressor, in
which case these three statistics will have different asymptotic
distributions.
This will be referred to as:
Regression
T:
2.2. Numerical
Regress yt on {l,t; .Yt_1,Y,-2,...,
Y,--6;
m,-,,
mr-2>...2
m~-6).
issues
Evaluation of the asymptotic distributions.
Sims, Stock and Watson (1986)
provided explicit expressions for the weak limits of the three statistics with
nonstandard
distributions.
These expressions - which are general enough to
apply to the statistics from both regressions
C and T - depend on certain
functionals
of Brownian motion over the unit interval and on the parameters
of the VAR. In general, all the parameters of the VAR in (1) will enter into
these expressions;
we henceforth refer to the vector of these parameters
as 8.
We evaluate these asymptotic distributions
numerically
by first generating and
storing sample path equivalents of these functionals,
computed using driftless
Gaussian
random walks with 1000 observations.
Given B (which is of course
known in this Monte Carlo experiment),
the percentiles
of the asymptotic
distributions
are approximated
by the percentiles of the empirical distribution
constructed
using 4000 draws of these previously computed functionals;
the
multiple draws of the random functionals
need to be computed only once. The
details of this procedure are discussed in Stock (1987) and Stock and Watson
(1987).
Evaluation
of the sampling distributions.
The Monte Carlo simulations
involve generating
n + 18 observations
of A y, and Am, according
to the
Gaussian
probability
model (1) where n = 100, 200, and 400. The 18 additional
observations
were judged sufficient to provide a stationary
initial
of the F( mI *,
, m,_,)-statistic
comes from recognizing that it
that tests whether the growth rate of money belongs in the income
the lagged money regressors
to be WI-,.
Am,~ ,,...,Am,
-5,
., mr_6) = F(Am,_, ,..., Am,__,).
4An alternative
interpretation
is equivalent to the F-statistic
equation.
That is, rewriting
F(m,-,...
L. Ljungqvist,
Multivariate
493
‘unit root’ distributions
distribution for a regression with n observations, given the low dependence
evident in the parameters in the appendix. The pseudo-observations were
cumulated and the levels regressions were run using n observations on the
dependent variable; the previous six observations were used as initial conditions.
All computations were done on a 16 MHz Compaq 80386/80387 desktop
computer using the GAUSS programming language. The total computation
time for the asymptotic distributions reported here was 2f minutes, exclusive
of the one-time computation of the multiple draws of the random functionals
of Brownian motion. The Monte Carlo simulations (4000 draws) required 25
hours.
3. Results
The finite sample and asymptotic distributions of the four statistics computed in regression C are presented in figs. 1-4; the corresponding distributions for regression T are shown in figs. 5-8. Selected percentiles of the
asymptotic distributions are tabulated in table 1. For purposes of comparison,
the ‘usual’ asymptotic distributions (that would apply were the regressors all
stationary) are also presented for the tsY,(ri and F(m,_,, . . . , m,_,)-statistics.
Focusing first on th! results for regression C, inspection of figs. l-4 and
table 1 indicate that n/3,,,(1) is sharply skewed and has a nonzero mean, both
in finite samples and in the limit. While this nonzero mean seems substantial,
20
l.~g~tld:
.____--
-
16
-
n-100
-
-
-
n-200
-
n-400
n-m
12
0
x10-
i
I
1
I
I
I
0
5
10
15
20
Fig. 1. Distributions
for regression
C: n&,,(1).
494
L. Ljungqvist, Multivariate
‘unit root’ distributions
.6
.5
Legend:
-_--_
-.
-
-
n-lOO
n-200
n-400
.‘1-
.3
-
-2
-1
0
1
2
3
4
Fig. 2. Distributions for regression C: fpvmclj.
Fig. 3. Distributions for regression C: Fv_,(m,_l,.
., m,-6).
495
L. Ljungqvist, Multivariate ‘unit root’ distributions
-----
n-200
n-400
n-
-
.6
.2
-0’
3
2
1
0
5
4
Fig. 4. Distributions for regression C: <v,m(m,_2,.
., WI-~)
Legend:
----_.
------n-400
,la102
/ /+y
-40
,
,
-20
0
“-100
n-200
;\.::_;____
7.0
Fig. 5. Distributions for regression T: n/$,,(l).
40
60
L. Ljungqvist,
496
Multivariate
‘unit roof’ distributions
/\
/
Legend:
\
/
---_-- ---
\
I
\
‘\
*loo
-
-
n-200
n-400
-
“-
-
Standard
normal
\
Fig. 6. Distributions
1
for regression
T: ta,,cl,.
.o
Legend:
_-_--*loll
--- n-200
---n400
--
Fig. 7. Distributions
for regression
T: Fv,,, (m,_ 1,.
II-
2
x6/6
Distribution
, m,_6)
L Ljungqvist, Multivariate ‘unit root’ distributions
491
Legend:
---__
- ---
.6
Fig. 8. Distributions for regression T: c,,,, ( M, _ 2,.
n-100
-- n-200
-n-400
1m,_6).
since 8,,(l)
converges at the rate n the bias is in fact small; for example, with
n = 100, the bias implied by the asymptotic distribution
is 0.0362. Comparing
the finite sample and limiting distributions
suggests that the convergence
of
n&,(l)
is rather slow, with substantial
differences in the left tail and center
even with n = 400. The fade- statistic inherits the positive mean of 8,,(l).
However, in contrast to p,,,(l), the convergence of the finite sample distributions of ta (i) to their limit is sufficiently fast that the asymptotic and n = 400
distributio&
are very close. Inspection of fig. 3 suggests that, like t, c1j, the
f&m the
distribution
of the F(m,_,, . . . , mr_6 )-statistic differs substantially
‘usual’ x26/6 distribution.
Again, the convergence of the finite sample distributions to the asymptotic
limit is fast, in the sense that the n = 400 and
asymptotic
distributions
are essentially the same. Finally, the finite sample
distributions
of the F( m,_2,. . . , m,_,)-statistic
are quite close to their x:/5
asymptotic
limit.
Including
time as a regressor evidently makes a substantial
difference in the
distributions
of some of these statistics. On the one hand, the asymptotic
distributions
seem to provide good approximations
for n = 400 (and in some
cases n = 200); on the other hand, the shapes of the nonstandard
asymptotic
distributions
differ sharply between regressions C and T. The distributions
of
&??(l) and t&_(l) from regression T both have a mean close to zero and
exhibit substantially
less skewness than in regression C. It is worth noting,
however, that the limiting distribution
of tS,mclj still has substantially
heavier
tails than the standard normal limit.
- 27.35
- 2.60
(-1.96)
0.22
(0.21)
0.17
- 33.57
- 3.07
((2.32)
0.17
(0.15)
0.11
&I(%*Y....m,H,)
&H(l)
r&Cl)
5%
Table I
25%
Percentile
distributions.a
50%
0.53
1.12
(0.58)
1.59
(- 0.67)
2.23
3.62
0.87
1.55
(0.89)
(0.00)
2.10
0.41
(0.37)
-1.77
(- 1.28)
~ 17.22
0.65
(0.58)
-- 0.95
( ~ 0.67)
- 8.67
1.00
(0.89)
(0.00)
0.00
0.01
for statistics estimated in regression T
0.32
0.77
(0.37)
1.09
(- 1.28)
1.32
statistics estimated in regression C
PO%
of asymptotic
1.48
(1.31)
0.99
(0.67)
9.01
1.33
2.11
(1.31)
2.61
(0.67)
5.56
15%
2.06
(1.77)
1.83
(1.28)
17.78
1.85
_______
2.68
(1.77)
3.08
(1.28)
7.78
90%
2.45
(2.10)
2.30
(1.64)
23.34
2.21
3.05
(2.10)
3.38
(1.64)
9.33
95%
2.75
(2.41)
2.68
(1.96)
28.51
2.56
~____
3.43
(2.41)
3.63
(1.96)
11.12
97.5%
3.33
(2.80)
3.11
(2.32)
35.24
3.00
3.84
(2.80)
3.95
(2.32)
12.82
998
____~
0.32
0.53
0.87
1.33
1.85
2.21
2.56
3.00
____.
Fv,m(m,_2r..
, m, .,)-statistic
has an asymptotic
x:/5 distribution;
the other statistics have
computing the nonstandard
asymptotic distributions
is described in the text. The percentiles in
obtained using Gaussian asymptotic theory: percentiles of a standard normal are given for the
F,,,(m,
,,
, m,_,)-statistic.
0.23
0.30
(0.27)
- 2.23
((1.64)
- 22.69
B. Percentiles
0.23
0.61
(0.27)
0.74
(~ 1.64)
0.86
A. Percentiles/or
aThe number of observations
is denoted by n. The
nonstandard
asymptotic distributions.
The method for
parentheses
correspond
to the ‘usual’ (incorrect) values
tS,_(,,-statistic.
and xi/6 percentiles are given for the
m,A
0.17
0.11
&Am,~,,‘~‘>m,-6)
Fy.m(mr-Z1...,
0.47
(0.21)
0.34
(0.15)
‘&Al)
m,A
0.43
0.37
(- 1.96)
-0.10
- 0.06
(-2.32)
&n(l)
L,(m,+t.....
2.5%
1%
Statistic
Percentiles
!?
2
rr.
E
z?.
P
-;
G
z
2.
L. Ljungqvist, Multivariate
499
‘unit root’ distributions
Table 2
Monte Carlo rejection probabilities for tests of level a based on the correct asymptotic criticai
values.”
n
1%
100
0.01
0.01
0.01
0.04
0.04
200
400
0.02
0.02
0.01
c
100
200
400
0.02
0.01
0.01
T
100
200
400
0.02
0.02
0.01
C
100
200
400
0.02
0.01
0.01
T
100
200
400
0.02
0.02
0.02
5%
10%
25%
50%
0.04
0.07
0.08
0.09
0.17
0.20
0.23
0.35
0.42
0.47
0.08
0.07
0.14
0.13
0.29
0.28
0.55
0.52
0.06
0.11
0.26
0.50
0.05
0.05
0.05
0.09
0.10
0.09
0.21
0.22
0.23
0.41
0.43
0.48
0.08
0.07
0.06
0.14
0.12
0.11
0.30
0.29
0.26
0.54
0.52
0.51
0.07
0.06
0.06
0.12
0.11
0.11
0.27
0.26
0.26
0.52
0.51
0.51
0.07
0.07
0.06
0.14
0.12
0.11
0.29
0.27
0.26
0.55
0.51
0.51
‘4. 'P,,Cl)
C
200
400
T
100
B. Fv,m(M,_l,...,m,-6)
c.
f3”.m(m,_2r...,
m,-6)
“The t-test is two-sided. Based on 4000 Monte Carlo simulations using data generated from
model (1) as described in the text. The correct asymptotic percentiles are taken from table 1 (for
regression C and regression T); nonstandard distribution theory was used to obtain the correct
critical values for the tests in panels A and B.
A basic question motivating this investigation is whether tests calculated
using the asymptotic critical values have the desired sizes in finite samples, To
this end, the sizes of tests (based on the correct critical values) using ta,,Cr,,
m1_6) are given in table 2. These results
m,-6) and F(m,_2,...,
F(m,_t,...,
suggest that for small samples the asymptotic approximation can be rather
unsatisfactory; for example, with n = 100, a two-sided t-test based on the
asymptotic 25% critical value would have a size of 17%.5 However, in most
cases the size is quite close to the level for n = 200; this is true in all cases for
n = 400.
‘The two-sided t-tests were performed using the square of the statistic and the corresponding
asymptotic critical value.
L. Ljungyuist, Multioariate ‘unrt root’ distrhutions
500
Table 3
Monte
Carlo
rejection
probabilities
for tests of level a based on the incorrect
asymptotic critical values.’
(Gaussian
and x’
)
a
n
1%
C
100
200
400
0.26
0.31
0.36
T
100
200
400
0.12
0.11
0.09
c
100
200
400
0.08
0.08
0.08
T
100
200
400
0.04
0.03
0.03
_____~
5%
10%
25%
50%
0.69
0.75
0.83
0.85
0.90
0.95
1.00
1.00
1.00
0.42
0.39
0.38
0.68
0.66
0.66
1.00
1.00
1.00
0.21
0.23
0.24
0.32
0.35
0.37
0.54
0.57
0.62
0.75
0.80
0.85
0.13
0.11
0.10
0.20
0.19
0.16
0.37
0.37
0.35
0.62
0.62
0.60
___
A. ‘P,,,Cl,
0.55
0.62
0.70
0.29
0.27
0.26
~__
B. ~.,,,(fi~,_L >..., fil,_,)
-___“The t-test is two-sided. Based on 4000 Monte Carlo simulations
using data generated from
model (1). 95% confidence intervals for these rejection probabilities
range from k 0.003 for the 1%
level tests to iO.014 for the 50% level tests. The (incorrect) critical values used to perform the
tests are the applicable values given in parentheses in table 1.
Finally, it is of some interest to consider what mistakes might be made were
the ‘usual’ Gaussian critical values used to perform the tests. To this end, the
sizes of tests calculated
using the tp,“,C1j- and F(m,_,, . , . , m,_,)-statistics,
based on the incorrect standard critical values, are presented in table 3. In
almost all cases the size is substantially
greater than the level; this is particularly true for tests based on regression C and on the tg,,,cl,-statistic.
4. Conclusions
It is important
to mention two cautionary
notes about interpreting
these
results too broadly.
First, this analysis has focused ona single probability
model. While this made it possible to proceed with the analysis, and while this
probability
model may be typical of models studied when analyzing
the
relation between money and income, there is no reason that the same quantitative conclusions
would obtain were we to analyze a different model. Second,
we have considered
a model that is linear, with known autoregressive
order,
and - most importantly
- in which there are exact unit roots. It would be
surprising
indeed were these assumptions
precisely to describe any extant
macroeconomic
time series, although they might be satisfactory
approxima-
L Ljungqvist, Multivariate
‘unit root’ distributions
501
tions in the context of certain linear prediction problems with stochastically
trending
variables.
Thus this investigation
has proceeded on grounds most
favorable
to the asymptotic
‘unit roots’ distribution
theory. In this light,
satisfactory
performance
of the asymptotic theory in this experiment is but a
minimal condition
for thinking that it will provide a satisfactory approximation in practice.
Viewed within the context of this exercise, however, our results suggest four
general conclusions.
First, the nonstandard
asymptotic distributions
of certain
estimators
can be sharply skewed and can have a nonzero mean. Second, the
rate of convergence
of the finite sample distributions
to their nonstandard
limits is fast, in the practical sense that the sizes of tests performed using series
of lengths typically found in empirical macroeconomic
research are close to
their asymptotic level. Third, substantial errors in inference can be made if the
‘usual’ critical values predicated on the Gaussian theory of stationary regressors are used in implementing
t- and F-tests that have nonstandard
limits.
These three observations
are consistent with earlier results both for univariate
‘unit roots’ distributions
and for the distributions
of estimators of cointegrating vectors. Finally, an additional lesson suggested by these results is that the
nonstandard
asymptotic (and finite sample) distributions
can be very sensitive
to seemingly
minor changes in the specification,
such as including
a time
trend.
Appendix
Table 4
Coefficients
Regressor
Constant
AY,-,
AY,+,
AY,-,
AY,-,
AY,-s
Am,_r
Amr-2
Am,-,
Am,-4
Am,+5
of probability
model (l).a
A y equation
Am equation
0.0015
0.0026
0.3919
0.0822
0.0469
0.0386
- 0.0679
0.0449
- 0.0104
- 0.0031
- 0.0258
0.0068
O.OOQO
0.0000
0.0000
0.0000
0.0000
0.2486
- 0.0407
0.1557
- 0.2040
0.1325
“These coefficients are the point estimates obtained by OLS using monthly industrial production growth and nominal Ml growth (detrended) as described in the text. The estimation period
was 1948:7 to 1985:12. with earlier observations
used for initial values.
502
L. Ljungqvist, Multivariate
‘unit root’ distributions
References
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